Calculate integral int_1^e 1/x((ln x)^2+1)dx?
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Luca B.
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You need to evaluate the given definite integral, hence, you should use the integration by substitution, such that:
`ln x = t => (1/x)dx = dt`
You need to change the limits of integration, such that:
`x = 1 => ln 1 = 0 => t = 0`
`x = e => ln e = 1 => t = 1`
Changing the variable and the limits of integration yields:
`int_0^1 (dt)/(t^2+1) = tan^(-1) t|_0^1`
Using the fudamental theorem of calculus, yields:
`int_0^1 (dt)/(t^2+1) = tan^(-1) 1 - tan^(-1) 0`
`int_0^1 (dt)/(t^2+1) = pi/4 - 0`
`int_0^1 (dt)/(t^2+1) = pi/4`
Hence, evaluating the given definite integral, using integration by substitution, yields `int_1^e 1/x((ln x)^2+1) dx = pi/4` .
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